Wave Theory and Free-Surfaces - A Study of the Motion of Oil Reservoir Fluids

This paper deals with the time-varying solutions of free-surface problems in porous media as an extension of the steady-state theory developed by M. King Hubbert. Examination of the equations shows that a marked similarity exists between these and those of the linear exact wave theory for the propagation of a disturbance at the interface of two superposed fluids. This purely formal analogy is extended so that solutions in hydrodynamic theory may be transformed quite simply to give solutions to similar problems in porous media. The auxiliary problem in water wave theory, namely the shallow water problem, is next considered. The resulting equation is found to exhibit a similar analogy to the Boussinesq equation or the Dupuit-Forchheimer theory of ground-water flow. On the basis of these analogies and the relationship between the theories of deep and shallow water waves, an analysis is made of the errors involved in the Dupuit approximation. A discussion on the application of the Dupuit approximation to model experiments indicates the importance of the size of the model on the interpretation of results. A method is suggested for analysing gravity flow problems by means of a combination of the Dupuit theory and the more exact theory. Introduction This paper is concerned with the study of the motion of an interface separating immiscible liquids of different density and viscosity in a porous medium. It is assumed that the flow of the fluids in question is governed by Darcy's law and that the presence of an interface is due to complete segregation under the action of gravitational forces. Hence in a multi-fluid system, whenever gravitational effects are of importance, interface or free-surface problems naturally present themselves. Strictly speaking, the term "free-surface" is usually applied to the hypothetical surface of demarcation between air and water in ground water studies. However, it is a convenient description and is applied quite generally to surfaces of separation as defined above, thus avoiding any confusion with the interfacial phenomena studied in capillarity theory.